\(\left(1-\frac{1}{a}\right)\left(1-\frac{1}{b}\right)=1-\frac{1}{a}-\frac{1}{b}+\frac{1}{ab}\le1-\frac{1}{2\sqrt{ab}}+\frac{1}{ab}\)

Ta có:\(4=a+b\ge2\sqrt{ab}\Rightarrow\sqrt{ab}\le2\Rightarrow\frac{1}{\sqrt{ab}}\ge2\)

Đặt \(x=\frac{1}{\sqrt{ab}}\ge2\Rightarrow A=1-\frac{a}{2}+a^2\) đạo hàm  phát ra ngay nè <3 

Cach khac: a+b>= 2 căn ab => 4 >= 2 căn ab => 4>= ab

\(A=\left(1-\frac{1}{a}\right).\left(1-\frac{1}{b}\right)=1-\frac{1}{a}-\frac{1}{b}+\frac{1}{ab}\)

Ta có: \(1-\frac{1}{a}-\frac{1}{b}+\frac{1}{ab}-\frac{1}{4}\)

\(=\frac{ab-4}{ab}-\frac{ab-4}{4ab}\left(a+b=4\right)\)

\(=\left(ab-4\right).\frac{3}{4}ab\le0\left(a,b>0;4\ge ab\right)\)

\(\Rightarrow A=1-\frac{1}{a}-\frac{1}{b}+\frac{1}{ab}\le\frac{1}{4}\)

Dấu = xr khi a=b=4/2=2

Vậy Max A=1/4 đạt tại a=b=2

Bài làm:

đkxđ: \(x,y\ge0;x\ne y\)

Ta có: \(\frac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}-\sqrt{y}\right)^2\)

\(=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)}{\sqrt{x}+\sqrt{y}}-x+2\sqrt{xy}-y\)

\(=x-\sqrt{xy}+y-x+2\sqrt{xy}-y\)

\(=\sqrt{xy}\)

\(\Leftrightarrow\frac{\left(\sqrt{x}\right)^3+\left(\sqrt{y}\right)^3}{\sqrt{x}+\sqrt{y}}-\left(x+y-2\sqrt{xy}\right)=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)}{\sqrt{x}+\sqrt{y}}-\left(x+y-2\sqrt{xy}\right)\)

\(\Leftrightarrow x-\sqrt{xy}+y-\left(x+y-2\sqrt{xy}\right)=\sqrt{xy}\)

Gọi m là độ dài một cạnh của hình vuông đó 

aabb là diện tích của hình vuông đó(\(1000\le\overline{aabb}\le9999\))

ta có aabb = m²

=> 1100a + 11b = m²

=> 11(100a + b) = m²

suy ra \(m^2⋮11\Rightarrow m⋮11\left(tcm\right)\)

ta có aabb = m²

và \(1000\le\overline{aabb}\le9999\)

nên \(32\le m\le99\)

thử lần lượt với 33;44;55;66;77;88;99

ta có 88 thỏa mãn 

vậy m = 88

khai triển ra thôi 

\(D=\left(2x-1\right)^2-2\left(2x-3\right)^2+4\)

\(=4x^2-4x+1-2\left(4x^2-12x+9\right)+4\)

\(=4x^2-4x+1-8x^2+24x-18+4\)

\(=-4x^2+20x-13\)

\(D=\left(2x-1\right)^2-2\left(2x-3\right)^2+4\)

\(=4x^2-4x+1-8x^2+24x-18+4\)

\(=-4x^2+20x-17+4=4x^2+20x-13\)

chùm hoa mới nở như một cô công chúa tí hon đẹp đẽ trào đời

cây đa tỏa bóng mát khắp sân trường

bạn mèo nhà em rất dễ thương

bố em là một bác sĩ

em rất ngoan

mùa hoa nở trên ngọn đồi vốn sác sơ , ko lá.

       hok tốt

1;như em bé mới dc xinh ra.

2;cho các bác nông dân nghỉ ngơi ở đó.

3;rất tinh nghịch hoặc nghịch ngợm .

4;là bác sĩ.

5;thích con mèo nhà em, bởi nó rát đg yêu.

I. Circle the correct answer.

Question 1: When I last ________ him, he was living in London.

    A. see    B. have seen    C. saw    D. seen

Question 2: Who .......... food in your family when your Mom is away?

    A. cooks    B. is cooking    C. has been cooking    D. cooked

Question 3: The train ...............half an hour ago.

    A. has been leaving    B. left    C. has left    D. was leaving

Question 4: Jack ................... the door. It looks wet.

    A. has just painted    B. paint    C. will have painted    D. painting

Question 5: My sister ............... for you since yesterday.

    A. is looking    B. was looking    C. has been looking    D. looked

Question 6: I ................ Texas State University now.

    A. am attending    B. attend    C. was attending    D. attended

Question 7: He has worked in this factory..............................

    A. ten years ago    B. Since ten years    C. for ten years ago    D. for ten years

Question 8: Where is John? – He .............. his car in the garage.

    A. repairs    B. is repairing    C. has repaired    D. repaired

Question 9: I love this film. I .................... it four or five times already.

    A. see    B. have seen    C. had seen    D. will see

Question 10: Have you visited any European countries? 

Yes. I .............. Spain and Italy two years ago.

    A. visited    B. have visited    C. had visited    D. will visit

Question 11: I ............ to him when he comes back home.

    A. will give    B. give    C. gave    D. am giving

Question 12: My father .................for Mr. Smith for six years.

    A. has worked    B. will work

    C. have been working        D. is working

Question 13: Where is he? I ............... for him since three o'clock!

    A. am waiting    B. have been waiting    C. was waiting    D. had been waiting

Question 14: I can't leave now. I ................... for an important telephone call.

    A. wait    B. am waiting    C. had been waiting    D. was waiting

Question 15: I have waited for you ..................

    A. since early morning B. since 9a.m    C. for two hours    D. All are correct

II. Fill  in each blank with since or for.

1. She's lived here _since_ 1975.

2. He's been in London _for_ five days.

3. They haven't been used _for_ many years.

4. She's been working _since_ nine this morning.

5.I haven't eaten _for_ hours.

6. It has been raining _for_ ages.

7. India has been independent _since_  1947.

8. They have been married _since_ October.

9. How many years is it _since_ you graduated from junior high school?

10. We have been waiting _for_ twenty minutes. 

III. Fill in each blank with   since/for/ yet/ already

1. Things have changed a lot………since……….our previous meeting.
2. It's been raining……for……….four o'clock.
3. Have you done your homework .......yet.................?
4. We've ...........already............ had dinner
5. I've been trying to fix this computer……for……….early this morning.
6. He hasn't had anything to eat…….for……….nearly a week.
7. It's three years……since……….he started learning German.
8. He's been collecting stamps……for……….the past twenty years.

9. Nobody has seen him……since……….last Friday.
10. It has been foggy………for…….some days.
11. He hasn't finished his work .........yet................
12. They've been living in France ……for……….eight years.
13. The pilots have been on strike…….for……….two months.

14. We've had terrible weather………since……. Saturday.

15. I've known Tom………since…….1990.
16. We have been waiting for the bus………for…….half an hour.
17. She hasn't lost a match………since…….April.
18. Things have changed……since……….she's become headmaster.
19. The police have been looking for him………for…….a month.
20. Our dog has been ill………for…….two days.

IV. Put the verbs into the present perfect tense.

1. The bill isn’t right. They (make)……have made….. a mistake.

2. Don’t you want to see this programme? It …has started………. (start).

3. I (turn)…have turned……. the heating on. It’ll soon get warm in here.

4. …Have…..they (pay)…paid….. money for your mother yet?

5. Someone (take)……has taken…………. my bicycle.

6. Wait for few minutes, please! I (finish)…have finished…………. my dinner.

7. …Have……you ever (eat)……eaten…….. Sushi?

8. She (not/come)……hasn't come……… here for a long time.

9. I (work)…have worked………... here for three years.

10. ……Have…… you ever ……been……..(be) in New York?

11. You (not/do) ……haven't done…….yor project yet, I suppose.

12. I (just/ see)…have just seen……. Andrew and he says he …has…..already (do)…done……. about half of the plan.

13. I …have……..just (decide)…decided…… to start working next week.

14. He (be)…has been………. at his computer for seven hours.

15. She (not/have) ……hasn't had………any fun a long time.

16. My father (not/ play)…hasn't played…….. any sport since last year.

17. I’d better have a shower. I (not/have)…haven't had……. one since Thursday.

18. I don’t live with my family now and we (not/see)…haven't seen………. each other for five years.

19. I…have… just (realize)…… realized……... that there are only four weeks to the end of term.

20. The train drivers (go)…have gone…… on strike and they stopped working at twelve o’clock.

V. The past simple or the present perfect?

1) The police......arrested..............two people early this morning. (arrest)

2) She …..has taugh……….in this school since she ……graduated…………from university. (teach/ graduate)

3) Dan.....has already bought.............two tablets this year. (already/ buy)

4) How many games...............has your team won................so far this season? (your team/ win)

5)......Did you visit..........the CN Tower when you......stayed..........in Toronto? (you/ visit - stay)

6).........Have you done...........your homework yet? – Yes, I........finished.........it an hour ago. (you/ do - finish)

7) There......Has been......an accident in High Street, so we have to use King Street to get to your school. (to be)

8) I......haven't seen.........Peter since I.....arrived..........last Tuesday. (not/ see - arrive)

9) Frank........got..........his bike in May. So he......has had.............it for 4 months. (get - have)

10) I'm sorry, I..........haven't written........earlier, but I..........have worked........a lot lately. (not/ write - work)

11) Peter....played........... football yesterday (play)

12) They......have cleaned........... the car. It looks new again. (clean)

13) Last year we.......went........... to Italy. (be)

14) John and Peggy......have just read................ the book. Now they can watch the film. (read)

15) I......met........... my friend two days ago. (meet)

16) We never……have never visited……….... another country before. (visit)

17) She...........bought.............. a new car in 2011. (buy)

18) I'm sorry, but I............have forgotten............... my homework. (forget)

19)......Did......you.........win......... the game of chess? (win)

20) The girls..........haven't eaten............. their lunch yet. (eat)

VI. Rewrite the following sentences.

1.She has never drunk whisky before.

This is the first time she ever drunk whisky before

2.He has never seen this film before.

This is the first time I haven't seen this film

3.This is the first time we’ve visited Ha Long Bay.

We have never visited Ha Long Bay before

4.The last time she visited me was 5 months ago.

She hasn't visited me for 5 months

5. I last had my hair cut when I left her.

I haven't cut my hair since i left her

6.It is a long time since we last met.

 We haven't met for a long time

I. Cirlce the correct answer.

Question 1: When I last ________ him, he was living in London.

A. see B. have seen C. saw D. seen

Question 2: Who .......... food in your family when your Mom is away?

A. cooks B. is cooking C. has been cooking D. cooked

Question 3: The train ...............half an hour ago.

A. has been leaving B. left C. has left D. was leaving

Question 4: Jack ................... the door. It looks wet.

A. has just painted B. paint C. will have painted D. painting

Question 5: My sister ............... for you since yesterday.

A. is looking B. was looking C. has been looking D. looked

Question 6: I ................ Texas State University now.

A. am attending B. attend C. was attending D. attended

Question 7: He has worked in this factory..............................

A. ten years ago B. Since ten years C. for ten years ago D. for ten years

Question 8: Where is John? – He .............. his car in the garage.

A. repairs B. is repairing C. has repaired D. repaired

Question 9: I love this film. I .................... it four or five times already.

A. see B. have seen C. had seen D. will see

Question 10: Have you visited any European countries?

Yes. I .............. Spain and Italy two years ago.

A. visited B. have visited C. had visited D. will visit

Question 11: I ............to him when he comes back home.

A. will give B. give C. gave D. am giving

Question 12: My father .................for Mr. Smith for six years.

A. has worked B. will work

C. have been working D. is working

Question 13: Where is he? I ............... for him since three o'clock!

A. am waiting B. have been waiting C. was waiting D. had been waiting

Question 14: I can't leave now. I ................... for an important telephone call.

A. wait B. am waiting C. had been waiting D. was waiting

Question 15: I have waited for you ..................

A. since early morning B. since 9a.m C. for two hours D. All are correct

II. Fill in each blank with since or for.

1. She's lived here __since_ 1975.
2. He's been in London _for__ five days.
3. They haven't been used _for__ many years.
4. She's been working _for__ nine this morning.
5. I haven't eaten _for__ hours.
6. It has been raining _for__ ages.
7. India has been independant __since_ 1947.
8. They have been married _since__ October.
9. How many years is it __since_ you graduated from junior high school?
10. We have been waiting _for__ twenty minutes.

Bài làm:

Ta có: \(1+\frac{1}{3}+\frac{1}{6}+...+\frac{1}{66}\)

\(=\frac{1}{1}+\frac{1}{1.3}+\frac{1}{3.2}+...+\frac{1}{11.6}\)

\(=\frac{1}{2}\left(\frac{1}{1.2}+\frac{1}{2.1.3}+\frac{1}{2.3.2}+...+\frac{1}{2.11.6}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{11.12}\right)\)

\(=\frac{1}{2}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{11}-\frac{1}{12}\right)\)

\(=\frac{1}{2}\left(1-\frac{1}{12}\right)\)

\(=\frac{1}{2}.\frac{11}{12}\)

\(=\frac{11}{24}\)

\(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}\)

\(=\frac{2}{2}+\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+\frac{2}{30}+...+\frac{2}{90}+\frac{2}{110}+\frac{2}{132}\)

\(=2\times\left(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+\frac{1}{4\times5}+\frac{1}{5\times6}+...+\frac{1}{9\times10}+\frac{1}{10\times11}+\frac{1}{11\times12}\right)\)

\(=2\times\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{10}-\frac{1}{11}+\frac{1}{11}-\frac{1}{12}\right)\)

\(=2\times\left(1-\frac{1}{12}\right)\)

\(=2\times\frac{11}{12}\)

\(=\frac{11}{6}\)

Xét \(\Delta BDC\)và \(\Delta EBC\)

ta có \(\widehat{BDC}=\widehat{EBC}\)(cùng phụ với \(\widehat{BED}\))

\(\widehat{DCB}=\widehat{BCE}=90^0\)

nên \(\Delta BDC\)đông dạng \(\Delta EBC\)(g-g)

dễ chứng minh \(\Delta BCD\approx\Delta EBD\left(g-g\right)\)

nên \(\frac{BD}{DE}=\frac{CD}{BD}\Rightarrow BD^2=CD\cdot DE\)

ABCD là hbh mà, làm sao